Abstract
For a domainU on a certaink-dimensional minimal submanifold ofS n orH n, we introduce a “modified volume”M(U) ofU and obtain an optimal isoperimetric inequality forU k k ω k M (D)k-1 ≤Vol(∂D)k, where ω k is the volume of the unit ball ofR k. Also, we prove that ifD is any domain on a minimal surface inS n+ (orH n, respectively), thenD satisfies an isoperimetric inequality2π A≤L 2+A2 (2π A≤L2−A2 respectively). Moreover, we show that ifU is ak-dimensional minimal submanifold ofH n, then(k−1) Vol(U)≤Vol(∂U).
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Choe, J., Gulliver, R. Isoperimetric inequalities on minimal submanifolds of space forms. Manuscripta Math 77, 169–189 (1992). https://doi.org/10.1007/BF02567052
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DOI: https://doi.org/10.1007/BF02567052